Lightly grease a medium baking dish.
Place the rolled oats in a bowl, add the milk, stir and leave to soak for 10 minutes.
Add the eggs, pinch of salt and vanilla extract to the soaked oats and mix well until smooth.
Grate the carrot finely and stir it into the oat mixture.
If the mixture is very thick, add 1–3 tbsp water, stirring until you have a thick but pourable batter.
Pour the mixture into the prepared baking dish and spread it into an even layer.
Core and thinly slice the apples, then arrange the slices on top of the oat mixture in one or two overlapping layers.
Sprinkle the apples with lemon juice and cinnamon.
Bake at 360°F (about 180°C) for 40 minutes, or until the top is golden and the centre is set.
Allow the bake to cool slightly before slicing and serving warm.